Percolation conjecture for the infinitely reinforced planar 2-out model

Let $\mathbb Z^d=(V,E)$ be the hypercubic lattice, and let $\mathcal E_\infty$ denote the set of edges that are reinforced in every round eventually, equivalently the edges reinforced infinitely often. An edge is called open when it belongs to $\mathcal E_\infty$.

Percolation conjecture for $d=k=2$. Assume that $d=k=2$. Then

$$\mathbb P\big(\mathcal E_\infty\text{ percolates}\big)=1.$$

This is the paper's principal conjecture for the limiting reinforcement model $\alpha=\infty$. The authors provide a finite-size criterion implying the claimed percolation, but do not prove that its hypothesis holds; the conjecture therefore remains open.